caa a22 4500 445804831 CHVBK 20180317145153.0 cr unu---uuuuu 170323e20110301xx s 000 0 eng 10.1007/s00453-010-9403-3 doi (NATIONALLICENCE)springer-10.1007/s00453-010-9403-3 Log-Linear Convergence and Divergence oftheScale-Invariant (1+1)-ES in Noisy Environments [Elektronische Daten] [Mohamed Jebalia, Anne Auger, Nikolaus Hansen] Noise is present in many real-world continuous optimization problems. Stochastic search algorithms such as Evolution Strategies (ESs) have been proposed as effective search methods in such contexts. In this paper, we provide a mathematical analysis of the convergence of a (1+1)-ES on unimodal spherical objective functions in the presence of noise. We prove for a multiplicative noise model that for a positive expected value of the noisy objective function, convergence or divergence happens depending on the infimum of the support of the noise. Moreover, we investigate convergence rates and show that log-linear convergence is preserved in presence of noise. This result is a strong theoretical foundation of the robustness of ESs with respect to noise. Springer Science+Business Media, LLC, 2010 Numerical optimization nationallicence Noisy optimization nationallicence Stochastic optimization algorithms nationallicence Evolution strategies nationallicence Convergence nationallicence Convergence rates nationallicence Markov chains nationallicence Borel-Cantelli lemma nationallicence Jebalia Mohamed TAO Team—INRIA Saclay-Île-de-France, LRI—Paris-Sud University, 91405, Orsay Cedex, France aut Auger Anne TAO Team—INRIA Saclay-Île-de-France, LRI—Paris-Sud University, 91405, Orsay Cedex, France aut Hansen Nikolaus TAO Team—INRIA Saclay-Île-de-France, LRI—Paris-Sud University, 91405, Orsay Cedex, France aut Algorithmica Springer-Verlag 59/3(2011-03-01), 425-460 0178-4617 59:3<425 2011 59 453 https://doi.org/10.1007/s00453-010-9403-3 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00453-010-9403-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Jebalia Mohamed TAO Team—INRIA Saclay-Île-de-France, LRI—Paris-Sud University, 91405, Orsay Cedex, France aut NATIONALLICENCE 700 1- Auger Anne TAO Team—INRIA Saclay-Île-de-France, LRI—Paris-Sud University, 91405, Orsay Cedex, France aut NATIONALLICENCE 700 1- Hansen Nikolaus TAO Team—INRIA Saclay-Île-de-France, LRI—Paris-Sud University, 91405, Orsay Cedex, France aut NATIONALLICENCE 773 0- Algorithmica Springer-Verlag 59/3(2011-03-01), 425-460 0178-4617 59:3<425 2011 59 453 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer